Coherence for tricategories via weak vertical composition Eugenia - - PowerPoint PPT Presentation

Coherence for tricategories via weak vertical composition

Eugenia Cheng

School of the Art Institute of Chicago

1.

Plan

Aim: show that tricategories with just weak vertical composition are “weak enough”

2.

Plan

Aim: show that tricategories with just weak vertical composition are “weak enough” . . .and thus shed light on the source of weakness in higher categories.

2.

Plan

Aim: show that tricategories with just weak vertical composition are “weak enough” . . .and thus shed light on the source of weakness in higher categories.

• 1. Overview: degeneracy and braidings

2.

Plan

Aim: show that tricategories with just weak vertical composition are “weak enough” . . .and thus shed light on the source of weakness in higher categories.

• 1. Overview: degeneracy and braidings
• 2. Warm-up: strictification via cliques

2.

Plan

Aim: show that tricategories with just weak vertical composition are “weak enough” . . .and thus shed light on the source of weakness in higher categories.

• 1. Overview: degeneracy and braidings
• 2. Warm-up: strictification via cliques
• 3. Construction

2.

Plan

Aim: show that tricategories with just weak vertical composition are “weak enough” . . .and thus shed light on the source of weakness in higher categories.

• 1. Overview: degeneracy and braidings
• 2. Warm-up: strictification via cliques
• 3. Construction
• 4. Equivalence.

2.

• 1. Overview: degeneracy

Test principle for weak n-categories Doubly degenerate 3-categories should “be” braided monoidal categories.

3.

• 1. Overview: degeneracy

Test principle for weak n-categories Doubly degenerate 3-categories should “be” braided monoidal categories. dd 3-category 0-cells 1-cells degenerate

• 2-cells

3-cells

3.

• 1. Overview: degeneracy

Test principle for weak n-categories Doubly degenerate 3-categories should “be” braided monoidal categories. dd 3-category 0-cells 1-cells degenerate

• 2-cells

3-cells braided monoidal category

• bjects

morphisms

3.

• 1. Overview: degeneracy

Test principle for weak n-categories Doubly degenerate 3-categories should “be” braided monoidal categories. dd 3-category 0-cells 1-cells degenerate

• 2-cells

3-cells braided monoidal category

• bjects

morphisms vertical composition ⊗ weak Eckmann–Hilton braiding

3.

• 1. Overview: braiding from weak Eckmann–Hilton

4.

• 1. Overview: braiding from weak Eckmann–Hilton

horizontal units ∼

4.

• 1. Overview: braiding from weak Eckmann–Hilton

horizontal units ∼ interchange ∼

4.

• 1. Overview: braiding from weak Eckmann–Hilton

horizontal units ∼ interchange ∼ vertical units ∼

4.

• 1. Overview: braiding from weak Eckmann–Hilton

horizontal units ∼ interchange ∼ vertical units ∼ vertical units ∼

4.

• 1. Overview: braiding from weak Eckmann–Hilton

horizontal units ∼ interchange ∼ vertical units ∼ vertical units ∼ interchange ∼

4.

• 1. Overview: braiding from weak Eckmann–Hilton

horizontal units ∼ interchange ∼ vertical units ∼ vertical units ∼ interchange ∼ horizontal units ∼

4.

• 1. Overview: braiding from weak Eckmann–Hilton

horizontal units ∼ interchange ∼ vertical units ∼ vertical units ∼ interchange ∼ horizontal units ∼

tricategory braided monoidal category

doubly degenerate

4.

• 1. Overview: braiding from weak Eckmann–Hilton

horizontal units ∼ interchange ∼ vertical units ∼ vertical units ∼ interchange ∼ horizontal units ∼

tricategory braided monoidal category

doubly degenerate

strict 3-category symmetric monoidal category

4.

• 1. Overview: braiding from weak Eckmann–Hilton

horizontal units ∼ interchange ∼ vertical units ∼ vertical units ∼ interchange ∼ horizontal units ∼

tricategory braided monoidal category

doubly degenerate

strict 3-category symmetric monoidal category gray “semi-strict” Gray area“semi- strict”

4.

• 1. Overview: flavours of semi-strictness

vertical horizontal units units interchange GPS strict strict weak JK strict weak strict C weak strict strict “Law of conservation of complicatedness”

5.

• 1. Overview: flavours of semi-strictness

vertical horizontal units units interchange subtlety GPS strict strict weak JK strict weak strict not doubly degenerate C weak strict strict need weak vertical associativity “Law of conservation of complicatedness”

5.

• 1. Overview: flavours of semi-strictness

vertical horizontal units units interchange subtlety GPS strict strict weak JK strict weak strict not doubly degenerate C weak strict strict need weak vertical associativity type of ⊗

• r type of functors

“Law of conservation of complicatedness”

5.

• 1. Overview: flavours of semi-strictness

vertical horizontal units units interchange subtlety GPS strict strict weak JK strict weak strict not doubly degenerate C weak strict strict need weak vertical associativity type of enrichment type of ⊗

• r type of functors

“Law of conservation of complicatedness”

5.

• 1. Overview: flavours of semi-strictness

vertical horizontal units units interchange subtlety GPS strict strict weak JK strict weak strict not doubly degenerate C weak strict strict need weak vertical associativity what we enrich in type of enrichment type of ⊗

• r type of functors

“Law of conservation of complicatedness”

5.

• 1. Overview: flavours of semi-strictness

vertical horizontal units units interchange subtlety GPS strict strict weak JK strict weak strict not doubly degenerate C weak strict strict need weak vertical associativity what we enrich in type of enrichment type of ⊗

• r type of functors

“Law of conservation of complicatedness” We enrich in (Bicats, ×):

• bicategories
• strict functors
• ordinary products
• strict enrichment

We write Bicats-Cat.

5.

• 1. Overview: overall aim

ddBicats-Cat wBrMonCat U

6.

• 1. Overview: overall aim

ddBicats-Cat wBrMonCat U Σ

• eventually: biequivalence
• first: U essentially surjective on 0-cells

6.

• 1. Overview: overall aim

ddBicats-Cat wBrMonCat U Σ

• eventually: biequivalence
• first: U essentially surjective on 0-cells

A 2-cells 3-cells vertical ◦ (weak) UA

• bjects

morphisms ⊗ (weak)

6.

• 1. Overview: overall aim

ddBicats-Cat wBrMonCat U Σ

• eventually: biequivalence
• first: U essentially surjective on 0-cells

A 2-cells 3-cells vertical ◦ (weak) UA

• bjects

morphisms ⊗ (weak) Weak Eckmann-Hilton: —braiding = = = = ∼ ∼

6.

• 1. Overview: overall aim

ddBicats-Cat wBrMonCat U Σ

• eventually: biequivalence
• first: U essentially surjective on 0-cells

A 2-cells 3-cells vertical ◦ (weak) UA

• bjects

morphisms ⊗ (weak) Weak Eckmann-Hilton: —braiding = = = = ∼ ∼ the only weak part

6.

• 1. Overview: essential surjectivity

We show ddBicats-Cat

U

wBrMonCat is essentially surjective on objects:

7.

• 1. Overview: essential surjectivity

We show ddBicats-Cat

U

wBrMonCat is essentially surjective on objects:

• start with a braided monoidal category B (with strict ⊗ wlog),

• 1. Overview: essential surjectivity

We show ddBicats-Cat

U

wBrMonCat is essentially surjective on objects:

• start with a braided monoidal category B (with strict ⊗ wlog),
• construct a ddBicats-category ΣB, and

• 1. Overview: essential surjectivity

We show ddBicats-Cat

U

wBrMonCat is essentially surjective on objects:

• start with a braided monoidal category B (with strict ⊗ wlog),
• construct a ddBicats-category ΣB, and
• a braided monoidal equivalence B

ΣB.

∼

7.

• 1. Overview: essential surjectivity

We show ddBicats-Cat

U

wBrMonCat is essentially surjective on objects:

• start with a braided monoidal category B (with strict ⊗ wlog),
• construct a ddBicats-category ΣB, and
• a braided monoidal equivalence B

ΣB.

∼

How do we get horizontal and vertical composition from ⊗?

7.

• 1. Overview: essential surjectivity

We show ddBicats-Cat

U

wBrMonCat is essentially surjective on objects:

• start with a braided monoidal category B (with strict ⊗ wlog),
• construct a ddBicats-category ΣB, and
• a braided monoidal equivalence B

ΣB.

∼

How do we get horizontal and vertical composition from ⊗? For tricategories: Put both as ⊗ and get interchange from the braiding γ

7.

• 1. Overview: essential surjectivity

We show ddBicats-Cat

U

wBrMonCat is essentially surjective on objects:

• start with a braided monoidal category B (with strict ⊗ wlog),
• construct a ddBicats-category ΣB, and
• a braided monoidal equivalence B

ΣB.

∼

How do we get horizontal and vertical composition from ⊗? For tricategories: Put both as ⊗ and get interchange from the braiding γ Issues:

• 1. We can’t have both compositions strict so they can’t both be ⊗.

• 1. Overview: essential surjectivity

We show ddBicats-Cat

U

wBrMonCat is essentially surjective on objects:

• start with a braided monoidal category B (with strict ⊗ wlog),
• construct a ddBicats-category ΣB, and
• a braided monoidal equivalence B

ΣB.

∼

How do we get horizontal and vertical composition from ⊗? For tricategories: Put both as ⊗ and get interchange from the braiding γ Issues:

• 1. We can’t have both compositions strict so they can’t both be ⊗.
• 2. We want interchange to be strict so it can’t be γ.

7.

• 1. Overview: essential surjectivity

We show ddBicats-Cat

U

wBrMonCat is essentially surjective on objects:

• start with a braided monoidal category B (with strict ⊗ wlog),
• construct a ddBicats-category ΣB, and
• a braided monoidal equivalence B

ΣB.

∼

How do we get horizontal and vertical composition from ⊗? For tricategories: Put both as ⊗ and get interchange from the braiding γ Issues:

• 1. We can’t have both compositions strict so they can’t both be ⊗.
• 2. We want interchange to be strict so it can’t be γ.

Solution: Do “weakification” for the vertical direction.

7.

• 2. Warm-up: strictification of weak monoidal categories

Coherence: Every weak monoidal category is monoidal equivalent to a strict one.

8.

• 2. Warm-up: strictification of weak monoidal categories

Coherence: Every weak monoidal category is monoidal equivalent to a strict one. Follows from: F1 is monoidal equivalent to the discrete (N, +, 0).

8.

• 2. Warm-up: strictification of weak monoidal categories

Coherence: Every weak monoidal category is monoidal equivalent to a strict one. Follows from: F1 is monoidal equivalent to the discrete (N, +, 0). So

• F1 splits into connected components Cn “bracketed words of length n”.

• 2. Warm-up: strictification of weak monoidal categories

Coherence: Every weak monoidal category is monoidal equivalent to a strict one. Follows from: F1 is monoidal equivalent to the discrete (N, +, 0). So

• F1 splits into connected components Cn “bracketed words of length n”.
• Each Cn ≃ 1 so all bracketings are uniquely isomorphic “all diagrams commute”.

8.

• 2. Warm-up: strictification of weak monoidal categories

Coherence: Every weak monoidal category is monoidal equivalent to a strict one. Follows from: F1 is monoidal equivalent to the discrete (N, +, 0). So

• F1 splits into connected components Cn “bracketed words of length n”.
• Each Cn ≃ 1 so all bracketings are uniquely isomorphic “all diagrams commute”.

Given a monoidal category M we construct stM “strictification”

8.

• 2. Warm-up: strictification of weak monoidal categories

Coherence: Every weak monoidal category is monoidal equivalent to a strict one. Follows from: F1 is monoidal equivalent to the discrete (N, +, 0). So

• F1 splits into connected components Cn “bracketed words of length n”.
• Each Cn ≃ 1 so all bracketings are uniquely isomorphic “all diagrams commute”.

Given a monoidal category M we construct stM “strictification”

• objects: words in the objects of M (unbracketed)

• 2. Warm-up: strictification of weak monoidal categories

Coherence: Every weak monoidal category is monoidal equivalent to a strict one. Follows from: F1 is monoidal equivalent to the discrete (N, +, 0). So

• F1 splits into connected components Cn “bracketed words of length n”.
• Each Cn ≃ 1 so all bracketings are uniquely isomorphic “all diagrams commute”.

Given a monoidal category M we construct stM “strictification”

• objects: words in the objects of M (unbracketed)
• morphisms: evaluate words in M then take morphisms from M.

• 2. Warm-up: strictification of weak monoidal categories

Coherence: Every weak monoidal category is monoidal equivalent to a strict one. Follows from: F1 is monoidal equivalent to the discrete (N, +, 0). So

• F1 splits into connected components Cn “bracketed words of length n”.
• Each Cn ≃ 1 so all bracketings are uniquely isomorphic “all diagrams commute”.

Given a monoidal category M we construct stM “strictification”

• objects: words in the objects of M (unbracketed)
• morphisms: evaluate words in M then take morphisms from M.

Question: How do we evaluate strict words in a weak monoidal category?

8.

• 2. Warm-up: strictification of weak monoidal categories

Coherence: Every weak monoidal category is monoidal equivalent to a strict one. Follows from: F1 is monoidal equivalent to the discrete (N, +, 0). So

• F1 splits into connected components Cn “bracketed words of length n”.
• Each Cn ≃ 1 so all bracketings are uniquely isomorphic “all diagrams commute”.

Given a monoidal category M we construct stM “strictification”

• objects: words in the objects of M (unbracketed)
• morphisms: evaluate words in M then take morphisms from M.

Question: How do we evaluate strict words in a weak monoidal category? Answer: Use cliques.

8.

• 2. Warm-up: strictification of weak monoidal categories

Thought experiment Suppose we’re trying to define morphisms abc y in stM.

9.

• 2. Warm-up: strictification of weak monoidal categories

Thought experiment Suppose we’re trying to define morphisms abc y in stM.

• I could take morphisms (a ⊗ b) ⊗ c

y.

9.

• 2. Warm-up: strictification of weak monoidal categories

Thought experiment Suppose we’re trying to define morphisms abc y in stM.

• I could take morphisms (a ⊗ b) ⊗ c

y.

• You could take morphisms a ⊗ (b ⊗ c)

y.

9.

• 2. Warm-up: strictification of weak monoidal categories

Thought experiment Suppose we’re trying to define morphisms abc y in stM.

• I could take morphisms (a ⊗ b) ⊗ c

y.

• You could take morphisms a ⊗ (b ⊗ c)

y.

• We could all take different ones by throwing in copies of I.

9.

• 2. Warm-up: strictification of weak monoidal categories

Thought experiment Suppose we’re trying to define morphisms abc y in stM.

• I could take morphisms (a ⊗ b) ⊗ c

y.

• You could take morphisms a ⊗ (b ⊗ c)

y.

• We could all take different ones by throwing in copies of I.

Key: they’re not really different.

9.

• 2. Warm-up: strictification of weak monoidal categories

Thought experiment Suppose we’re trying to define morphisms abc y in stM.

• I could take morphisms (a ⊗ b) ⊗ c

y.

• You could take morphisms a ⊗ (b ⊗ c)

y.

• We could all take different ones by throwing in copies of I.

Key: they’re not really different. All bracketings of abc are uniquely isomorphic via coherence constraints.

9.

• 2. Warm-up: strictification of weak monoidal categories

Thought experiment Suppose we’re trying to define morphisms abc y in stM.

• I could take morphisms (a ⊗ b) ⊗ c

y.

• You could take morphisms a ⊗ (b ⊗ c)

y.

• We could all take different ones by throwing in copies of I.

Key: they’re not really different. All bracketings of abc are uniquely isomorphic via coherence constraints. We just have to agree that we’ve got the same morphism if they only differ by that unique isomorphism:

9.

• 2. Warm-up: strictification of weak monoidal categories

Thought experiment Suppose we’re trying to define morphisms abc y in stM.

• I could take morphisms (a ⊗ b) ⊗ c

y.

• You could take morphisms a ⊗ (b ⊗ c)

y.

• We could all take different ones by throwing in copies of I.

Key: they’re not really different. All bracketings of abc are uniquely isomorphic via coherence constraints. We just have to agree that we’ve got the same morphism if they only differ by that unique isomorphism: (a ⊗ b) ⊗ c (a′ ⊗ b′) ⊗ c

(f ⊗ g) ⊗ h

• 2. Warm-up: strictification of weak monoidal categories

Thought experiment Suppose we’re trying to define morphisms abc y in stM.

• I could take morphisms (a ⊗ b) ⊗ c

y.

• You could take morphisms a ⊗ (b ⊗ c)

y.

• We could all take different ones by throwing in copies of I.

Key: they’re not really different. All bracketings of abc are uniquely isomorphic via coherence constraints. We just have to agree that we’ve got the same morphism if they only differ by that unique isomorphism: (a ⊗ b) ⊗ c (a′ ⊗ b′) ⊗ c

(f ⊗ g) ⊗ h

a ⊗ (b ⊗ c) a′ ⊗ (b′ ⊗ c′)

f ⊗ (g ⊗ h)

• 2. Warm-up: strictification of weak monoidal categories

Thought experiment Suppose we’re trying to define morphisms abc y in stM.

• I could take morphisms (a ⊗ b) ⊗ c

y.

• You could take morphisms a ⊗ (b ⊗ c)

y.

• We could all take different ones by throwing in copies of I.

Key: they’re not really different. All bracketings of abc are uniquely isomorphic via coherence constraints. We just have to agree that we’ve got the same morphism if they only differ by that unique isomorphism: (a ⊗ b) ⊗ c (a′ ⊗ b′) ⊗ c

(f ⊗ g) ⊗ h

a ⊗ (b ⊗ c) a′ ⊗ (b′ ⊗ c′)

f ⊗ (g ⊗ h) ∼ ∼

9.

• 2. Warm-up: cliques

Idea A clique is essentially a collection of objects and unique isomorphisms between them.

10.

• 2. Warm-up: cliques

Idea A clique is essentially a collection of objects and unique isomorphisms between them. · · · · ·

• 2. Warm-up: cliques

Idea A clique is essentially a collection of objects and unique isomorphisms between them. · · · · · Precisely/technically A clique in a category C is a functor J C where J ≃ 1

10.

• 2. Warm-up: cliques

Idea A clique is essentially a collection of objects and unique isomorphisms between them. · · · · · Precisely/technically A clique in a category C is a functor J C where J ≃ 1 — clique objects are indexed by J but can repeat.

10.

• 2. Warm-up: cliques

Idea A clique is essentially a collection of objects and unique isomorphisms between them. · · · · · Precisely/technically A clique in a category C is a functor J C where J ≃ 1 — clique objects are indexed by J but can repeat. The unique isomorphisms are called connecting isomorphisms.

10.

• 2. Warm-up: cliques

Idea A clique is essentially a collection of objects and unique isomorphisms between them. · · · · · Precisely/technically A clique in a category C is a functor J C where J ≃ 1 — clique objects are indexed by J but can repeat. The unique isomorphisms are called connecting isomorphisms. A clique map x y is a system of morphisms from each object of x to each object of y making everything commute.

10.

• 2. Warm-up: cliques

Idea A clique is essentially a collection of objects and unique isomorphisms between them. · · · · · Precisely/technically A clique in a category C is a functor J C where J ≃ 1 — clique objects are indexed by J but can repeat. The unique isomorphisms are called connecting isomorphisms. A clique map x y is a system of morphisms from each object of x to each object of y making everything commute. We only have to specify one component to know what all the others are.

10.

• 2. Warm-up: cliques

Idea A clique is essentially a collection of objects and unique isomorphisms between them. · · · · · · · · Precisely/technically A clique in a category C is a functor J C where J ≃ 1 — clique objects are indexed by J but can repeat. The unique isomorphisms are called connecting isomorphisms. A clique map x y is a system of morphisms from each object of x to each object of y making everything commute. We only have to specify one component to know what all the others are.

10.

• 2. Warm-up: cliques

Idea A clique is essentially a collection of objects and unique isomorphisms between them. · · · · · · · · Precisely/technically A clique in a category C is a functor J C where J ≃ 1 — clique objects are indexed by J but can repeat. The unique isomorphisms are called connecting isomorphisms. A clique map x y is a system of morphisms from each object of x to each object of y making everything commute. We only have to specify one component to know what all the others are.

10.

• 2. Warm-up: strictification of weak monoidal categories

Example There is a clique in M of all bracketings of abc (evaluated)

11.

• 2. Warm-up: strictification of weak monoidal categories

Example There is a clique in M of all bracketings of abc (evaluated) (a ⊗ b) ⊗ c a ⊗ (b ⊗ c)

∼

• 2. Warm-up: strictification of weak monoidal categories

Example There is a clique in M of all bracketings of abc (evaluated) (a ⊗ b) ⊗ c a ⊗ (b ⊗ c)

∼

(a′ ⊗ b′) ⊗ c a′ ⊗ (b′ ⊗ c′)

∼

• 2. Warm-up: strictification of weak monoidal categories

Example There is a clique in M of all bracketings of abc (evaluated) (a ⊗ b) ⊗ c a ⊗ (b ⊗ c)

∼

(a′ ⊗ b′) ⊗ c a′ ⊗ (b′ ⊗ c′)

∼ (f ⊗ g) ⊗ h

• 2. Warm-up: strictification of weak monoidal categories

Example There is a clique in M of all bracketings of abc (evaluated) (a ⊗ b) ⊗ c a ⊗ (b ⊗ c)

∼

(a′ ⊗ b′) ⊗ c a′ ⊗ (b′ ⊗ c′)

∼ (f ⊗ g) ⊗ h f ⊗ (g ⊗ h)

• ther components of the

same clique map

11.

• 2. Warm-up: strictification of weak monoidal categories

Example There is a clique in M of all bracketings of abc (evaluated) (a ⊗ b) ⊗ c a ⊗ (b ⊗ c)

∼

(a′ ⊗ b′) ⊗ c a′ ⊗ (b′ ⊗ c′)

∼ (f ⊗ g) ⊗ h f ⊗ (g ⊗ h)

• (

( f ⊗ g ) ⊗ h )

• ther components of the

same clique map

11.

• 2. Warm-up: strictification of weak monoidal categories

Example There is a clique in M of all bracketings of abc (evaluated) (a ⊗ b) ⊗ c a ⊗ (b ⊗ c)

∼

(a′ ⊗ b′) ⊗ c a′ ⊗ (b′ ⊗ c′)

∼ (f ⊗ g) ⊗ h f ⊗ (g ⊗ h) ((f ⊗ g) ⊗ h) ◦

− 1

• ther components of the

same clique map

11.

• 2. Warm-up: strictification of weak monoidal categories

Example There is a clique in M of all bracketings of abc (evaluated) (a ⊗ b) ⊗ c a ⊗ (b ⊗ c)

∼

(a′ ⊗ b′) ⊗ c a′ ⊗ (b′ ⊗ c′)

∼ (f ⊗ g) ⊗ h f ⊗ (g ⊗ h) ((f ⊗ g) ⊗ h) ◦

− 1

• ther components of the

same clique map We can compose components that don’t look composable via connecting isos (a ⊗ b) ⊗ c (a′ ⊗ b′) ⊗ c′ a′ ⊗ (b′ ⊗ c′) a′′ ⊗ (b′′ ⊗ c′′)

11.

• 2. Warm-up: strictification of weak monoidal categories

Example There is a clique in M of all bracketings of abc (evaluated) (a ⊗ b) ⊗ c a ⊗ (b ⊗ c)

∼

(a′ ⊗ b′) ⊗ c a′ ⊗ (b′ ⊗ c′)

∼ (f ⊗ g) ⊗ h f ⊗ (g ⊗ h) ((f ⊗ g) ⊗ h) ◦

− 1

• ther components of the

same clique map We can compose components that don’t look composable via connecting isos (a ⊗ b) ⊗ c a ⊗ (b ⊗ c) (a′ ⊗ b′) ⊗ c′ a′ ⊗ (b′ ⊗ c′) (a′′ ⊗ b′′) ⊗ c′′ a′′ ⊗ (b′′ ⊗ c′′)

11.

• 2. Warm-up: dots in boxes

12.

• 2. Warm-up: dots in boxes

Coherence for braided monoidal categories relates F1 to the braid category.

12.

• 2. Warm-up: dots in boxes

Coherence for braided monoidal categories relates F1 to the braid category. Braids come from configurations of points in R2, and paths.

12.

• 2. Warm-up: dots in boxes

Coherence for braided monoidal categories relates F1 to the braid category. Braids come from configurations of points in R2, and paths. We use configurations of points in the interior of I 2 · · · ·

12.

• 2. Warm-up: dots in boxes

Coherence for braided monoidal categories relates F1 to the braid category. Braids come from configurations of points in R2, and paths. We use configurations of points in the interior of I 2 · · · ·

• Vertical composition

· · · · · · · ·

12.

• 2. Warm-up: dots in boxes

Coherence for braided monoidal categories relates F1 to the braid category. Braids come from configurations of points in R2, and paths. We use configurations of points in the interior of I 2 · · · ·

• Vertical composition

· · · · · · · ·

• Horizontal composition

· · · · · · · ·

12.

• 2. Warm-up: dots in boxes

Coherence for braided monoidal categories relates F1 to the braid category. Braids come from configurations of points in R2, and paths. We use configurations of points in the interior of I 2 · · · ·

• Vertical composition

· · · · · · · ·

• Horizontal composition

· · · · · · · · Problem: Both weak.

12.

• 2. Warm-up: dots in boxes

Coherence for braided monoidal categories relates F1 to the braid category. Braids come from configurations of points in R2, and paths. We use configurations of points in the interior of I 2 · · · ·

• Vertical composition

· · · · · · · ·

• Horizontal composition

· · · · · · · · Problem: Both weak. Solution: Take “horizontal path” classes — paths that do not change any y coordinate · · · ·

• 2. Warm-up: dots in boxes

Coherence for braided monoidal categories relates F1 to the braid category. Braids come from configurations of points in R2, and paths. We use configurations of points in the interior of I 2 · · · ·

• Vertical composition

· · · · · · · ·

• Horizontal composition

· · · · · · · · Problem: Both weak. Solution: Take “horizontal path” classes — paths that do not change any y coordinate · · · · “strictification in the horizontal direction”

12.

• 3. Construction of ΣB

Aim: construct a ddBicats-category ΣB from a braided monoidal category B

13.

• 3. Construction of ΣB

Aim: construct a ddBicats-category ΣB from a braided monoidal category B Objects: horizontal path classes of points in I 2, labelled by objects of B

a1 . . . ak

· · · · ·

13.

• 3. Construction of ΣB

Aim: construct a ddBicats-category ΣB from a braided monoidal category B Objects: horizontal path classes of points in I 2, labelled by objects of B

a1 . . . ak

· · · · · Morphisms (cf strictification): start by evaluating the configuration as a word

13.

• 3. Construction of ΣB

Aim: construct a ddBicats-category ΣB from a braided monoidal category B Objects: horizontal path classes of points in I 2, labelled by objects of B

a1 . . . ak

· · · · · Morphisms (cf strictification): start by evaluating the configuration as a word

• similarity: there are many different ways to do so
• difference: they are not uniquely isomorphic

“not all diagrams commute”

13.

• 3. Construction of ΣB

Aim: construct a ddBicats-category ΣB from a braided monoidal category B Objects: horizontal path classes of points in I 2, labelled by objects of B

a1 . . . ak

· · · · · Morphisms (cf strictification): start by evaluating the configuration as a word

• similarity: there are many different ways to do so
• difference: they are not uniquely isomorphic

“not all diagrams commute” ·a ·b a ⊗ b

13.

• 3. Construction of ΣB

Aim: construct a ddBicats-category ΣB from a braided monoidal category B Objects: horizontal path classes of points in I 2, labelled by objects of B

a1 . . . ak

· · · · · Morphisms (cf strictification): start by evaluating the configuration as a word

• similarity: there are many different ways to do so
• difference: they are not uniquely isomorphic

“not all diagrams commute” ·a ·b a ⊗ b ·

a

·

b

?

• 3. Construction of ΣB

Aim: construct a ddBicats-category ΣB from a braided monoidal category B Objects: horizontal path classes of points in I 2, labelled by objects of B

a1 . . . ak

· · · · · Morphisms (cf strictification): start by evaluating the configuration as a word

• similarity: there are many different ways to do so
• difference: they are not uniquely isomorphic

“not all diagrams commute” ·a ·b a ⊗ b ·

a

·

b

?

clockwise a ⊗ b anti-clockwise b ⊗ a

13.

• 3. Construction of ΣB

Aim: construct a ddBicats-category ΣB from a braided monoidal category B Objects: horizontal path classes of points in I 2, labelled by objects of B

a1 . . . ak

· · · · · Morphisms (cf strictification): start by evaluating the configuration as a word

• similarity: there are many different ways to do so
• difference: they are not uniquely isomorphic

“not all diagrams commute” ·a ·b a ⊗ b ·

a

·

b

?

clockwise a ⊗ b anti-clockwise b ⊗ a There are many isomorphisms connecting these eg

13.

• 3. Construction of ΣB

Aim: construct a ddBicats-category ΣB from a braided monoidal category B Objects: horizontal path classes of points in I 2, labelled by objects of B

a1 . . . ak

· · · · · Morphisms (cf strictification): start by evaluating the configuration as a word

• similarity: there are many different ways to do so
• difference: they are not uniquely isomorphic

“not all diagrams commute” ·a ·b a ⊗ b ·

a

·

b

?

clockwise a ⊗ b anti-clockwise b ⊗ a There are many isomorphisms connecting these eg Solution: remember the journey, not just the destination.

13.

• 3. Construction of ΣB
• The free braided monoidal category embeds vertically:

·a ·b

a ⊗ b

·a ·b ·c

a ⊗ b ⊗ c

· · ·

14.

• 3. Construction of ΣB
• The free braided monoidal category embeds vertically:

·a ·b

a ⊗ b

·a ·b ·c

a ⊗ b ⊗ c

· · ·

• We “flatten” our configuration to a canonical vertical one

and remember what braid we used to do it. · · · · · · · · “flattening braid”

• 3. Construction of ΣB
• The free braided monoidal category embeds vertically:

·a ·b

a ⊗ b

·a ·b ·c

a ⊗ b ⊗ c

· · ·

• We “flatten” our configuration to a canonical vertical one

and remember what braid we used to do it. · · · · · · · · “flattening braid” · · · · clockwise a ⊗ b α

• 3. Construction of ΣB
• The free braided monoidal category embeds vertically:

·a ·b

a ⊗ b

·a ·b ·c

a ⊗ b ⊗ c

· · ·

• We “flatten” our configuration to a canonical vertical one

and remember what braid we used to do it. · · · · · · · · “flattening braid” · · · · clockwise a ⊗ b α · · · · anti-clockwise b ⊗ a β

• 3. Construction of ΣB
• The free braided monoidal category embeds vertically:

·a ·b

a ⊗ b

·a ·b ·c

a ⊗ b ⊗ c

· · ·

• We “flatten” our configuration to a canonical vertical one

and remember what braid we used to do it. · · · · · · · · “flattening braid” · · · · clockwise a ⊗ b α · · · · anti-clockwise b ⊗ a β · · · · hilarious b ⊗ a ξ

• 3. Construction of ΣB
• The free braided monoidal category embeds vertically:

·a ·b

a ⊗ b

·a ·b ·c

a ⊗ b ⊗ c

· · ·

• We “flatten” our configuration to a canonical vertical one

and remember what braid we used to do it. · · · · · · · · “flattening braid” · · · · clockwise a ⊗ b α · · · · anti-clockwise b ⊗ a β · · · · hilarious b ⊗ a ξ ! !

• 3. Construction of ΣB
• The free braided monoidal category embeds vertically:

·a ·b

a ⊗ b

·a ·b ·c

a ⊗ b ⊗ c

· · ·

• We “flatten” our configuration to a canonical vertical one

and remember what braid we used to do it. · · · · · · · · “flattening braid” · · · · clockwise a ⊗ b α · · · · anti-clockwise b ⊗ a β · · · · hilarious b ⊗ a ξ ! ! · · · · · ·

α−1 β connecting isomorphism γ

14.

• 3. Construction of ΣB: morphisms

stB ΣB Objects: strings of

• bjects of B

configurations of points labelled by

• bjects of B

15.

• 3. Construction of ΣB: morphisms

stB ΣB Objects: strings of

• bjects of B

configurations of points labelled by

• bjects of B

Morphisms:

• find associated clique in FB
• f bracketed words in B
• f flattened configurations

with flattening braids

15.

• 3. Construction of ΣB: morphisms

stB ΣB Objects: strings of

• bjects of B

configurations of points labelled by

• bjects of B

Morphisms:

• find associated clique in FB
• f bracketed words in B
• f flattened configurations

with flattening braids

• evaluate it as a clique in B and take clique maps in B

• 3. Construction of ΣB: morphisms

stB ΣB Objects: strings of

• bjects of B

configurations of points labelled by

• bjects of B

Morphisms:

• find associated clique in FB
• f bracketed words in B
• f flattened configurations

with flattening braids

• evaluate it as a clique in B and take clique maps in B

these are the morphisms of ΣB

15.

• 3. Construction of ΣB: morphisms

stB ΣB Objects: strings of

• bjects of B

configurations of points labelled by

• bjects of B

Morphisms:

• find associated clique in FB
• f bracketed words in B
• f flattened configurations

with flattening braids

• evaluate it as a clique in B and take clique maps in B

these are the morphisms of ΣB

• morphisms represent the same clique map if they differ only by

coherence constraints the correct factorising braid

15.

• 3. Construction of ΣB: example morphism

stB ΣB abc abc

?

16.

• 3. Construction of ΣB: example morphism

stB ΣB abc abc

?

rep by (ab)c (ab)c

id

16.

• 3. Construction of ΣB: example morphism

stB ΣB abc abc

?

rep by (ab)c (ab)c

id

• r

(ab)c a(bc)

16.

• 3. Construction of ΣB: example morphism

stB ΣB abc abc

?

rep by (ab)c (ab)c

id

• r

(ab)c a(bc)

id

16.

• 3. Construction of ΣB: example morphism

stB ΣB abc abc

?

rep by (ab)c (ab)c

id

• r

(ab)c a(bc)

id

16.

• 3. Construction of ΣB: example morphism

stB ΣB abc abc

?

rep by (ab)c (ab)c

id

• r

(ab)c a(bc)

id

— same clique map

16.

• 3. Construction of ΣB: example morphism

stB ΣB abc abc

?

rep by (ab)c (ab)c

id

• r

(ab)c a(bc)

id

— same clique map ·

a

·

b

·

a

·

b ?

16.

• 3. Construction of ΣB: example morphism

stB ΣB abc abc

?

rep by (ab)c (ab)c

id

• r

(ab)c a(bc)

id

— same clique map ·

a

·

b

·

a

·

b ?

· · · · a ⊗ b

• 3. Construction of ΣB: example morphism

stB ΣB abc abc

?

rep by (ab)c (ab)c

id

• r

(ab)c a(bc)

id

— same clique map ·

a

·

b

·

a

·

b ?

· · · · a ⊗ b · · · · b ⊗ a

• 3. Construction of ΣB: example morphism

stB ΣB abc abc

?

rep by (ab)c (ab)c

id

• r

(ab)c a(bc)

id

— same clique map ·

a

·

b

·

a

·

b ?

· · · · a ⊗ b · · · · b ⊗ a

γ

• 3. Construction of ΣB: example morphism

stB ΣB abc abc

?

rep by (ab)c (ab)c

id

• r

(ab)c a(bc)

id

— same clique map ·

a

·

b

·

a

·

b ?

· · · · a ⊗ b · · · · b ⊗ a

γ

· · a ⊗ b · · a ⊗ b

id

• 3. Construction of ΣB: example morphism

stB ΣB abc abc

?

rep by (ab)c (ab)c

id

• r

(ab)c a(bc)

id

— same clique map ·

a

·

b

·

a

·

b ?

· · · · a ⊗ b · · · · b ⊗ a

γ

· · a ⊗ b · · a ⊗ b

id id γ−1

— same clique map

16.

• 3. Construction of ΣB: example morphism

stB ΣB abc abc

?

rep by (ab)c (ab)c

id

• r

(ab)c a(bc)

id

— same clique map ·

a

·

b

·

a

·

b ?

· · · · a ⊗ b · · · · b ⊗ a

γ

· · a ⊗ b · · a ⊗ b

id id γ−1

— same clique map NB: identity in ΣB can be represented by a non-identity in B and vice versa

16.

• 3. Construction of ΣB: technically

Write O for the objects of B, FO for the free braided monoidal category on O.

17.

• 3. Construction of ΣB: technically

Write O for the objects of B, FO for the free braided monoidal category on O. We have functors Π1C(I 2, O) FO

F ∼

• 3. Construction of ΣB: technically

Write O for the objects of B, FO for the free braided monoidal category on O. We have functors Π1C(I 2, O) FO

F ∼

B

G eval

17.

• 3. Construction of ΣB: technically

Write O for the objects of B, FO for the free braided monoidal category on O. We have functors Π1C(I 2, O) FO

F ∼

B

G eval

inducing functors on clique categories

• Π1C(I 2, O)
• FO
• B

F ∗ G!

17.

• 3. Construction of ΣB: technically

Write O for the objects of B, FO for the free braided monoidal category on O. We have functors Π1C(I 2, O) FO

F ∼

B

G eval

inducing functors on clique categories

• Π1C(I 2, O)
• FO
• B

F ∗ G!

ΣB is defined by

• objects: horizontal path cliques of Π1C(I 2, O)
• morphisms:

ΣB(X, Y ) := B(G!F ∗X, G!F ∗Y )

17.

• 3. Construction of ΣB: “horizontal and vertical composition are both ⊗”

18.

• 3. Construction of ΣB: “horizontal and vertical composition are both ⊗”

Given a

f

a′ and b

g

b

′

• for any components f and g respectively, take f ⊗ g

18.

• 3. Construction of ΣB: “horizontal and vertical composition are both ⊗”

Given a

f

a′ and b

g

b

′

• for any components f and g respectively, take f ⊗ g
• we need to specify what component of the clique map it is

i.e. what flattening braids it refers to

18.

• 3. Construction of ΣB: “horizontal and vertical composition are both ⊗”

Given a

f

a′ and b

g

b

′

• for any components f and g respectively, take f ⊗ g
• we need to specify what component of the clique map it is

i.e. what flattening braids it refers to

• we know which flattening braids are referred to by f and g

18.

• 3. Construction of ΣB: “horizontal and vertical composition are both ⊗”

Given a

f

a′ and b

g

b

′

• for any components f and g respectively, take f ⊗ g
• we need to specify what component of the clique map it is

i.e. what flattening braids it refers to

• we know which flattening braids are referred to by f and g

Vertical composition: stack braids vertically

· · · · ····

· · · · · ·

18.

• 3. Construction of ΣB: “horizontal and vertical composition are both ⊗”

Given a

f

a′ and b

g

b

′

• for any components f and g respectively, take f ⊗ g
• we need to specify what component of the clique map it is

i.e. what flattening braids it refers to

• we know which flattening braids are referred to by f and g

Vertical composition: stack braids vertically

· · · · ····

· · · · · · Horizonal composition: stack braids horizontally. . . · · · · · · · · · · · · · ·

• 3. Construction of ΣB: “horizontal and vertical composition are both ⊗”

Given a

f

a′ and b

g

b

′

• for any components f and g respectively, take f ⊗ g
• we need to specify what component of the clique map it is

i.e. what flattening braids it refers to

• we know which flattening braids are referred to by f and g

Vertical composition: stack braids vertically

· · · · ····

· · · · · · Horizonal composition: stack braids horizontally. . . · · · · · · · · · · · · · · · · · · · · · . . .and twist

18.

• 3. Construction of ΣB: interchange

19.

• 3. Construction of ΣB: interchange

On objects:

c d a b c d a b c d a b

= =

19.

• 3. Construction of ΣB: interchange

On objects:

c d a b c d a b c d a b

= = On morphisms: by the method above we get different representatives

· · · ·

• 3. Construction of ΣB: interchange

On objects:

c d a b c d a b c d a b

= = On morphisms: by the method above we get different representatives

· · · ·

· · · ·

• 3. Construction of ΣB: interchange

On objects:

c d a b c d a b c d a b

= = On morphisms: by the method above we get different representatives

· · · ·

· · · · · · · · · ·

• 3. Construction of ΣB: interchange

On objects:

c d a b c d a b c d a b

= = On morphisms: by the method above we get different representatives

· · · ·

· · · · · · · · · · · · · ·

19.

• 3. Construction of ΣB: interchange

On objects:

c d a b c d a b c d a b

= = On morphisms: by the method above we get different representatives

· · · ·

· · · · · · · · · · · · · · · · · ·

19.

• 3. Construction of ΣB: interchange

On objects:

c d a b c d a b c d a b

= = On morphisms: by the method above we get different representatives

· · · ·

· · · · · · · · · · · · · · · · · · · · · ·

19.

• 3. Construction of ΣB: interchange

On objects:

c d a b c d a b c d a b

= = On morphisms: by the method above we get different representatives

· · · ·

· · · · · · · · · · · · · · · · · · · · · ·

connecting iso braid

19.

• 3. Construction of ΣB: interchange

On objects:

c d a b c d a b c d a b

= = On morphisms: by the method above we get different representatives

· · · ·

· · · · · · · · · · · · · · · · · · · · · ·

connecting iso braid

a ⊗ b ⊗ c ⊗ d a′ ⊗ b′ ⊗ c′ ⊗ d ′

f ⊗ g ⊗ h ⊗ i

• 3. Construction of ΣB: interchange

On objects:

c d a b c d a b c d a b

= = On morphisms: by the method above we get different representatives

· · · ·

· · · · · · · · · · · · · · · · · · · · · ·

connecting iso braid

a ⊗ b ⊗ c ⊗ d a′ ⊗ b′ ⊗ c′ ⊗ d ′

f ⊗ g ⊗ h ⊗ i

a ⊗ c ⊗ b ⊗ d a′ ⊗ c′ ⊗ b′ ⊗ d ′

f ⊗ h ⊗ g ⊗ i

• 3. Construction of ΣB: interchange

On objects:

c d a b c d a b c d a b

= = On morphisms: by the method above we get different representatives

· · · ·

· · · · · · · · · · · · · · · · · · · · · ·

connecting iso braid

a ⊗ b ⊗ c ⊗ d a′ ⊗ b′ ⊗ c′ ⊗ d ′

f ⊗ g ⊗ h ⊗ i

a ⊗ c ⊗ b ⊗ d a′ ⊗ c′ ⊗ b′ ⊗ d ′

f ⊗ h ⊗ g ⊗ i 1 ⊗ γ ⊗ 1 1 ⊗ γ ⊗ 1

— same clique map

19.

• 3. Construction of ΣB: interchange

On objects:

c d a b c d a b c d a b

= = On morphisms: by the method above we get different representatives

· · · ·

· · · · · · · · · · · · · · · · · · · · · ·

connecting iso braid

a ⊗ b ⊗ c ⊗ d a′ ⊗ b′ ⊗ c′ ⊗ d ′

f ⊗ g ⊗ h ⊗ i

a ⊗ c ⊗ b ⊗ d a′ ⊗ c′ ⊗ b′ ⊗ d ′

f ⊗ h ⊗ g ⊗ i 1 ⊗ γ ⊗ 1 1 ⊗ γ ⊗ 1

— same clique map Interchange is strict but still comes from the braiding.

19.

• 4. Braided monoidal equivalence B

∼ UΣB

20.

• 4. Braided monoidal equivalence B

∼ UΣB

• 1. Define functor
• 2. Equivalence of categories
• 3. Monoidal equivalence

20.

• 4. Braided monoidal equivalence B

∼ UΣB

• 1. Define functor

a

a

a b clique map represented by a b

• 2. Equivalence of categories
• 3. Monoidal equivalence

20.

• 4. Braided monoidal equivalence B

∼ UΣB

• 1. Define functor

a

a

a b clique map represented by a b

• 2. Equivalence of categories
• full and faithful by construction
• 3. Monoidal equivalence

20.

• 4. Braided monoidal equivalence B

∼ UΣB

• 1. Define functor

a

a

a b clique map represented by a b

• 2. Equivalence of categories
• full and faithful by construction
• essentially surjective on objects:

Given · · · ·

a1 . . . ak

∈ UΣB

• 3. Monoidal equivalence

20.

• 4. Braided monoidal equivalence B

∼ UΣB

• 1. Define functor

a

a

a b clique map represented by a b

• 2. Equivalence of categories
• full and faithful by construction
• essentially surjective on objects:

Given · · · ·

a1 . . . ak

∈ UΣB a1 ⊗ · · · ⊗ ak ∈ B

a1 ⊗ · · · ⊗ ak

• 3. Monoidal equivalence

20.

• 4. Braided monoidal equivalence B

∼ UΣB

• 1. Define functor

a

a

a b clique map represented by a b

• 2. Equivalence of categories
• full and faithful by construction
• essentially surjective on objects:

Given · · · ·

a1 . . . ak

∈ UΣB a1 ⊗ · · · ⊗ ak ∈ B

a1 ⊗ · · · ⊗ ak

and we have · · · ·

a1 . . . ak ak ⊗ · · · ⊗ ak ∼

• 3. Monoidal equivalence

20.

• 4. Braided monoidal equivalence B

∼ UΣB

• 1. Define functor

a

a

a b clique map represented by a b

• 2. Equivalence of categories
• full and faithful by construction
• essentially surjective on objects:

Given · · · ·

a1 . . . ak

∈ UΣB a1 ⊗ · · · ⊗ ak ∈ B

a1 ⊗ · · · ⊗ ak

and we have · · · ·

a1 . . . ak ak ⊗ · · · ⊗ ak ∼

• 3. Monoidal equivalence

This isomorphism is a clique map represented by an identity.

20.

• 4. Braided monoidal equivalence B

∼ UΣB

• 1. Define functor

a

a

a b clique map represented by a b

• 2. Equivalence of categories
• full and faithful by construction
• essentially surjective on objects:

Given · · · ·

a1 . . . ak

∈ UΣB a1 ⊗ · · · ⊗ ak ∈ B

a1 ⊗ · · · ⊗ ak

and we have · · · ·

a1 . . . ak ak ⊗ · · · ⊗ ak ∼

• 3. Monoidal equivalence

Need ·a ·b

a ⊗ b ∼

This isomorphism is a clique map represented by an identity.

20.

• 4. Braided monoidal equivalence B

∼ UΣB

• 4. Braided monoidal equivalence

21.

• 4. Braided monoidal equivalence B

∼ UΣB

• 4. Braided monoidal equivalence

Need ·a ·b

a ⊗ b rep by id

·b ·a

braiding from EH in ΣB b ⊗ a rep by id rep by γ from B

21.

• 4. Braided monoidal equivalence B

∼ UΣB

• 4. Braided monoidal equivalence

Need ·a ·b

a ⊗ b rep by id

·b ·a

braiding from EH in ΣB b ⊗ a rep by id rep by γ from B

— commuting diagram of clique maps

21.

• 4. Braided monoidal equivalence B

∼ UΣB

• 4. Braided monoidal equivalence

Need ·a ·b

a ⊗ b rep by id

·b ·a

braiding from EH in ΣB b ⊗ a rep by id rep by γ from B

— commuting diagram of clique maps · · · · · · · · · · · · · · · · · · · · · · · · EH in ΣB:

• 4. Braided monoidal equivalence B

∼ UΣB

• 4. Braided monoidal equivalence

Need ·a ·b

a ⊗ b rep by id

·b ·a

braiding from EH in ΣB b ⊗ a rep by id rep by γ from B

— commuting diagram of clique maps · · · · · · · · · · · · · · · · · · · · · · · · EH in ΣB: clique maps: id

• 4. Braided monoidal equivalence B

∼ UΣB

• 4. Braided monoidal equivalence

Need ·a ·b

a ⊗ b rep by id

·b ·a

braiding from EH in ΣB b ⊗ a rep by id rep by γ from B

— commuting diagram of clique maps · · · · · · · · · · · · · · · · · · · · · · · · EH in ΣB: clique maps: id id

• 4. Braided monoidal equivalence B

∼ UΣB

• 4. Braided monoidal equivalence

Need ·a ·b

a ⊗ b rep by id

·b ·a

braiding from EH in ΣB b ⊗ a rep by id rep by γ from B

— commuting diagram of clique maps · · · · · · · · · · · · · · · · · · · · · · · · EH in ΣB: clique maps: id id ∼

weak vertical units

• 4. Braided monoidal equivalence B

∼ UΣB

• 4. Braided monoidal equivalence

Need ·a ·b

a ⊗ b rep by id

·b ·a

braiding from EH in ΣB b ⊗ a rep by id rep by γ from B

— commuting diagram of clique maps · · · · · · · · · · · · · · · · · · · · · · · · EH in ΣB: clique maps: id id ∼

weak vertical units

id

• 4. Braided monoidal equivalence B

∼ UΣB

• 4. Braided monoidal equivalence

Need ·a ·b

a ⊗ b rep by id

·b ·a

braiding from EH in ΣB b ⊗ a rep by id rep by γ from B

— commuting diagram of clique maps · · · · · · · · · · · · · · · · · · · · · · · · EH in ΣB: clique maps: id id ∼

weak vertical units

id id

• 4. Braided monoidal equivalence B

∼ UΣB

• 4. Braided monoidal equivalence

Need ·a ·b

a ⊗ b rep by id

·b ·a

braiding from EH in ΣB b ⊗ a rep by id rep by γ from B

— commuting diagram of clique maps · · · · · · · · · · · · · · · · · · · · · · · · EH in ΣB: clique maps: id id ∼

weak vertical units

id id reps id

• 4. Braided monoidal equivalence B

∼ UΣB

• 4. Braided monoidal equivalence

Need ·a ·b

a ⊗ b rep by id

·b ·a

braiding from EH in ΣB b ⊗ a rep by id rep by γ from B

— commuting diagram of clique maps · · · · · · · · · · · · · · · · · · · · · · · · EH in ΣB: clique maps: id id ∼

weak vertical units

id id reps id

braiding

∼

• 4. Braided monoidal equivalence B

∼ UΣB

• 4. Braided monoidal equivalence

Need ·a ·b

a ⊗ b rep by id

·b ·a

braiding from EH in ΣB b ⊗ a rep by id rep by γ from B

— commuting diagram of clique maps · · · · · · · · · · · · · · · · · · · · · · · · EH in ΣB: clique maps: id id ∼

weak vertical units

id id reps id

braiding

∼ id

• 4. Braided monoidal equivalence B

∼ UΣB

• 4. Braided monoidal equivalence

Need ·a ·b

a ⊗ b rep by id

·b ·a

braiding from EH in ΣB b ⊗ a rep by id rep by γ from B

— commuting diagram of clique maps · · · · · · · · · · · · · · · · · · · · · · · · EH in ΣB: clique maps: id id ∼

weak vertical units

id id reps id

braiding

∼ id id id

• 4. Braided monoidal equivalence B

∼ UΣB

• 4. Braided monoidal equivalence

Need ·a ·b

a ⊗ b rep by id

·b ·a

braiding from EH in ΣB b ⊗ a rep by id rep by γ from B

— commuting diagram of clique maps · · · · · · · · · · · · · · · · · · · · · · · · EH in ΣB: clique maps: id id ∼

weak vertical units

id id reps id

braiding

∼ id id id main idea

21.

Conclusion Main ideas

• Putting points in boxes gives us enough control.
• Using cliques exchanges the roles of units and interchange.

22.

Conclusion Main ideas

• Putting points in boxes gives us enough control.
• Using cliques exchanges the roles of units and interchange.

Main results

• We have Σ on 0-cells.

ddBicats-Cat wBrMonCat U Σ

• It is biessentially surjective on objects.

22.

Conclusion Main ideas

• Putting points in boxes gives us enough control.
• Using cliques exchanges the roles of units and interchange.

Main results

• We have Σ on 0-cells.

ddBicats-Cat wBrMonCat U Σ

• It is biessentially surjective on objects.

Weak vertical composition is enough to produce braidings.

22.

• 5. Further work

Done but no space in talk:

• Define weak functors of ddBicats-categories using abstract EH (CT18).
• Assemble these into a 2-category with icon-like transformations.
• Extend Σ to a pseudo-functor of 2-categories.
• Show that we have a biequivalence of 2-categories.
• Analogous results for Trimble 3-categories.

Future:

• Rotate and get weak horizontal composition and strict vertical.
• Produce free doubly-degenerate structures by composing adjunctions.
• The non-degenerate case.
• Higher dimensions.

23.